Pressure runaway in a 2D plane channel with permeable walls submitted to pressure-dependent suction

Abstract

A leaking duct carries a flow that is a characteristic of several applications. Devices for cross-flow microfiltration are composed of a duct, the walls of which are semi-permeable membranes. In subsurface irrigation, the walls of watering pipes can be of porous clay, whereas the watering hoses are riddled with holes or made of porous material, in surface drip irrigation. In these applications, the first approach consists of assuming that the flow concerns a pure fluid (as in a microfiltration system operating at a very low species concentration), and that the wall’s leakage depends only on the local pressure difference between both inner and outer sides of the wall. Actually, only the situation with fixed uniform permeation at the walls is clearly understood. In the case of a 2D plane channel, the configuration is referred to as Berman flow. Berman theory predicts the existence of a critical transverse Reynolds number, denoted R t i s o (with R t i s o ≈ 1.31 ), at which both viscous and inertial effects compensate to maintain the flow isobaric. A recent contribution has extended the Berman theory to non-uniform leakage depending linearly on local pressure, and focused on the situation where R i n , the transverse Reynolds number at the inlet, is smaller than R t i s o . In contrast, the present work investigates the case R i n > R t i s o , for which the leakage is responsible for a pressure enhancement inside the channel. Now, if the transpiration increases with the local pressure, we are faced with a possible runaway in pressure. This phenomenon (generally unwanted in the applications) is here analysed. The particular study of the dead-end channel configuration shows that multiple flows can be found.